I teach a course on Applied Linear Algebra, intended for Engineers, where the final project is always to give a presentation on an application of linear algebra in the student's field of study. Since I myself am not an engineer, I often learn new things here, but there is the risk that my level of understanding is no better than an undergraduate engineer.

With this caveat, one of the common topics my students talk about is image identication. You have, say, $M$ reference photos of people's faces, each of them $K \times K$ grey-scale pixels and positioned similarly in the camera frame. You get a new photo and you want to identify which face it is most like.

The naive thing to due is to treat each of the $M$ images as a vector with $K^2$ entries, and find the reference photo in $K^2$ dimensional space which is closest to your photo. The trouble is that you will be thrown off by random noise introduced by the photography process, and you will get a face which is closest to your photo in the directions of random variation, and not in the directions that faces naturally vary.

A better idea is to use SVD on the $M \times K^2$ matrix whose columns are your reference photos, to find a lower dimensional plane $V$ in $K^2$-dimensional space, such that all of your reference photos are close to $V$. Then project all the reference photos, and the input photo, onto $V$, before finding the closest one. This way, the output photo will be close to the input photo in the ways in which faces vary, while the random noise will be projected away.

This is called the eigenfaces method. It can of course apply to images other than faces, such as optical character recognition.